We provide a sketch of the methods used in this manuscript. The set of primes \(\Sigma _0\)  is chosen to consist of exactly those primes at which \(E_1\)  or \(E_2\)  have bad reduction. One may choose a larger set, however, results are optimal for this set. Greenberg and Vatsal in [1]} consider the \(\Sigma _0\) -imprimitive Selmer group over the cyclotomic \(\mathbb {Z}_p\) -extension, which we denote by \(\operatorname{Sel}^{\Sigma _0}(E/\mathbb {Q}^{cyc})\)  (see section \(\ref {section2}\)  for the definition). Set \(f_{E_i,\Sigma _0}^{alg}(T)\)  for the characteristic series of \(\operatorname{Sel}^{\Sigma _0}(E_i/\mathbb {Q}^{cyc})\)  and denote by \(\mu _{E_i,\Sigma _0}^{alg}\)  and \(\lambda _{E_i,\Sigma _0}^{alg}\)  the \(\mu \)  and \(\lambda \) -invariants of \(f_{E_i,\Sigma _0}^{alg}(T)\)  respectively. Greenberg and Vatsal show that
\(\mu _{E_1,\Sigma _0}^{alg}=0\Leftrightarrow \mu _{E_2,\Sigma _0}^{alg}=0\text{, and if }\mu _{E_1,\Sigma _0}^{alg}=\mu _{E_2,\Sigma _0}^{alg}=0\text{, then }\lambda _{E_1,\Sigma _0}^{alg}=\lambda _{E_2,\Sigma _0}^{alg}\) 
